Galois sections and p-adic period mappings

Let K be a number field not containing a CM subfield. For any smooth projective curve Y/K of genus >= 2, we prove that the image of the "Selmer" part of Grothendieck's section set inside the Kv-rational points Y (Kv) is finite for every finite place v. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.